Computers & Geosciences 50 (2013) 84–94

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Computers & Geosciences

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Evolutionary-based approaches for determining the deviatoricstress of calcareous sands

Habib Shahnazari a,n, Mohammad A. Tutunchian a, Reza Rezvani a, Fatemeh Valizadeh b

a School of Civil Engineering, Iran University of Science and Technology, Narmak, PO Box 16765-163, Tehran, Iranb School of Engineering, Tarbiat Modares University, PO Box 14115-179, Tehran, Iran

a r t i c l e i n f o

Article history:

Received 3 April 2012

Received in revised form

6 July 2012

Accepted 9 July 2012Available online 20 July 2012

Keywords:

Calcareous sands

Dataset

Modeling

Genetic programming

Triaxial experiments

04/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.cageo.2012.07.006

esponding author. Tel.: þ98 21 73913128.

ail addresses: hshahnazari@iust.ac.ir (H. Shah

ust.ac.ir (M.A. Tutunchian), r_rezvani@iust.ac

eh@modares.ac.ir (F. Valizadeh).

a b s t r a c t

Many hydrocarbon reservoirs are located near oceans which are covered by calcareous deposits. These

sediments consist mainly of the remains of marine plants or animals, so calcareous soils can have a

wide variety of engineering properties. Due to their local expansion and considerable differences from

terrigenous soils, the evaluation of engineering behaviors of calcareous sediments has been a major

concern for geotechnical engineers in recent years. Deviatoric stress is one of the most important

parameters directly affecting important shearing characteristics of soils. In this study, a dataset of

experimental triaxial tests was gathered from two sources. First, the data of previous experimental

studies from the literature were gathered. Then, a series of triaxial tests was performed on calcareous

sands of the Persian Gulf to develop the dataset. This work resulted in a large database of experimental

results on the maximum deviatoric stress of different calcareous sands. To demonstrate the capabilities

of evolutionary-based approaches in modeling the deviatoric stress of calcareous sands, two promising

variants of genetic programming (GP), multigene genetic programming (MGP) and gene expression

programming (GEP), were applied to propose new predictive models. The models’ input parameters

were the physical and in-situ condition properties of soil and the output was the maximum deviatoric

stress (i.e., the axial-deviator stress). The results of statistical analyses indicated the robustness of these

models, and a parametric study was also conducted for further verification of the models, in which the

resulting trends were consistent with the results of the experimental study. Finally, the proposed

models were further simplified by applying a practical geotechnical correlation.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In 1968, a very large pile displacement occurred during a piledriving project on the Lavan Petroleum platform in the PersianGulf. This was the first issue with calcareous deposits observed byengineers (McClelland (1988)). Since then, many similar con-struction problems in calcareous deposits, such as low resistanceagainst pile driving, low bearing capacity, and high particlecrushing potential have been reported by different researchers(Wang et al., 2011; Brandes, 2011).

Carbonate sediments have been observed in temperate andtropical areas near important hydrocarbon industries and petro-chemical reserves (e.g., the Persian Gulf of Iran, Hawaiian Islands,Puerto Rico, Republic of Ireland, and Australia). Approximately 40%of ocean beds are covered by such carbonate deposits (Holmes,1978). Carbonate sediments are mainly formed by the skeletal

ll rights reserved.

nazari),

.ir (R. Rezvani),

remains of marine organisms. Therefore, a wide variety of engineer-ing properties can be found in these soils due to different locationsand fauna that contribute to their formation. Therefore, it has beencomplicated to predict the geotechnical properties and mechanicalbehavior of calcareous soils (Kaggwa et al., 1988; Kaggwa, 1988;King and Lodge, 1988; Airey, 1993; Hassanlourad et al., 2008, 2011;Dehnavi et al., 2010; Wang et al., 2011). Research over the past 40years has also shown fundamental differences in mechanical char-acteristics of calcareous soils compared to terrigenous noncarbo-nated soils (Datta et al., 1982; Coop, 1990; Celestino and Mitchell,1983; Brandes, 2011; Rezvani et al., 2011).

Due to the large range of physical properties in calcareous soils,the estimation of engineering parameters such as deviatoric stress isassociated with some degree of uncertainty. However, performingexperimental studies is sometimes necessary. Experimental investi-gations also have drawbacks such as the difficulty of obtainingundisturbed samples, the use of expensive and time-consumingtesting procedures, and the need for advanced laboratory testingequipments such as a triaxial apparatus. Data mining and patternrecognition techniques can help to solve this problem. Thesetechniques can be used to develop numerical correlations based

Nomenclature

GP genetic programmingMGP multigene genetic programmingGEP gene expression programmingemax maximum void ratioemin minimum void ratioemax-emin or De void ratio rangeCu uniformity coefficientCc coefficient of curvature

jmax maximum friction angleR2 coefficient of determinationRMSE root mean square errorMAE mean absolute errorq deviatoric stressqmax maximum deviatoric stressCD consolidated drained triaxial tests03 effective confining pressures01 effective axial stress

H. Shahnazari et al. / Computers & Geosciences 50 (2013) 84–94 85

on experimental results (from the literature) and propose predictiveformulae for practical engineering applications.

Modeling the full stress–strain behavior of calcareous soils canbe useful and aid in determining the maximum deviatoric stress.Some studies such as Javadi and Rezania (2009a, 2009b), Cabalarand Cevik (2011), and Johari et al. (2011) have used soft comput-ing techniques to simulate triaxial tests results. This studyfocused on modeling the maximum deviatoric stress of calcareoussands due to unavailability of full stress-strain behaviors of somedata obtained from the literature. The maximum deviatoric stressis an important engineering parameter for the evaluation ofstress-strain theories of soils. For instance, it is one of the chiefmodeling parameters of the ‘‘Hyperbolic Model,’’ which is a well-known constitutive model for non-linear elastic geo-materials.The maximum deviatoric stress is also necessary for calculation ofthe maximum friction angle.

The purpose of this study was to develop evolutionary-basedmodels for prediction of the maximum deviatoric stress. A datasetof experimental triaxial tests on various types of carbonate sandsfrom different locations of the world was gathered from theliterature. To extend the data, an experimental investigationwas performed on two different calcareous sands. By calculatingthe statistical properties of the extended database, the databasewas divided into two statistically consistent subsets, training andtesting. The training subset was used to develop novel predictivemodels for estimation of the maximum deviatoric stress ofcalcareous sands in drained conditions. The testing subset wasemployed to measure the accuracy of the developed models.

In this study, two promising variants of genetic programming(GP) were considered to model the maximum deviatoric stress,multigene genetic programming (MGP) and gene expressionprogramming (GEP). Statistical analyses indicated the robustnessof the proposed models in terms of the coefficient of correlation(R2), root mean square error (RMSE) and mean absolute error(MAE). A parametric study was also conducted to investigate therobustness of the new MGP- and GEP-based models. Finally,simpler formulae (with one less input parameter) were obtainedby considering a geotechnical relationship between two of theinput parameters. All the developed models can be used forestimating the maximum friction angle.

Fig. 1. Definition of maximum deviatoric stress for evolutionary-based modelings.

2. Database

The database for this study was created in two phases. The firstphase focused on gathering previously published triaxial testresults on calcareous sands. In order to extend the database, inthe second phase, an experimental study was carried out on twocalcareous sands of different origins in the Persian Gulf, HormuzIsland and Bushehr Port. The collected database consisted of 90experimental triaxial results, 66 triaxial test results from previous

studies and 24 test results from the experimental portion of thecurrent research.

The physical properties of calcareous sands, including max-imum void ratio (emax), minimum void ratio (emin), averageparticle size (D50) and uniformity coefficient (Cu), and fieldcondition properties including after consolidation relative density(Drac) and effective confining pressure (s03), were used as inputparameters, and the maximum deviatoric stress (qmax) in drainedconditions was used as the output.

It should be noted that the definition of qmax is important forthe measurement of the maximum deviatoric stress from adeviatoric stress–strain curve. If the observed qmax occurrs inlower than or equal to 20% strain, then qmax is defined as the peakpoint of the deviatoric stress–strain curve (see curve A in Fig. 1). Ifthe observed maximum deviatoric stress occurrs in strains higherthan 20%, then qmax is considered to be the value of deviatoricstress at 20% strain (see curve B in Fig. 1).

In some triaxial tests, the strength of the sample was increasedcontinuously during loading to high strain values. Many research-ers have reported that a very large strain is needed to reachmaximum strength in calcareous sands (Brandes, 2011; Coopet al., 2004; Sharma and Ismail, 2006). It has been stated bySharma and Ismail (2006) that more than 20% axial strain isrequired in triaxial testing for carbonate soil to mobilize itsmaximum strength. However, at high axial strain values (morethan 20%) the soil sample experiences large deformations, and theresults at such a high strain are unreliable for engineering design.Therefore as mentioned above, a deviatoric stress value

H. Shahnazari et al. / Computers & Geosciences 50 (2013) 84–9486

corresponding to 20% axial strain was considered to specify qmax

of such samples.

2.1. Data from the literature

Calcareous deposits exhibit a variety of behaviors due to thedifferent organisms that make up their formations. Therefore, theyhave a wide range of physical properties. This study gathered datafrom previous studies performed on calcareous soils from differentgeographical regions (the Persian Gulf, Hawaii, Ireland and England).Morioka (1999) performed 14 static triaxial tests on Ewa Plains sandat the University of Hawaii and Hyodo et al. (1994) conducted4 tests on the Dogs Bay sand of Ireland at the University ofYamaguchi. The results of 48 triaxial tests on the carbonate sandsof the Persian Gulf (including Kish, Hormuz and Tonbak sands) andEngland (Rock Beach sand) reported by Hassanlourad et al., 2008 atIran University of Science and Technology were also employed inthis study. The Scanning Electron Microscope (SEM) photographs ofthe Rock and Kish sands are shown in Figs. 2 and 3, respectively, anddemonstrate the shape and some of the fauna that make up thecalcareous formations. They also present inter-particle voids ofcalcareous sands that play an important role in the behavior ofthese deposits.

2.2. Data from experimental research

A series of monotonic isotropically consolidated drained (CD)triaxial tests was conducted to extend the database. To study theeffects of density and confining pressure, the reconstitutedsamples were prepared at different initial densities and consoli-dated under confining pressures ranging from 100 to 600 kPa. Thetests were terminated when the axial strain reached to 20%. Thediameter and height of the specimens were 70 and 140 mm,

Fig. 2. Microscopic image of the Rock

respectively. The samples were prepared by the air pluviationmethod and the saturation procedure was performed in threesteps, including passing carbon dioxide (CO2) from the specimens,exuding de-aired water into the samples, and applying a backpressure of 200 kPa. Carbon dioxide gas (CO2) was used toexhaust the pore air trapped between sand particles and increasethe quality of saturation in the specimen. The pressure of the gaswas low to avoid heterogeneity in the sample. More details aboutthe monotonic triaxial testing procedure can be found in Rezvaniet al. (2011).

Two calcareous sands, obtained from Hormuz Island andBushehr Port on the northern coast of the Persian Gulf, wereused in triaxial experiments. Both sands were formed by thin-walled mollusk and echinoderm plate fragments and thick-walled foraminifera. Table 1 provides a brief summary ofphysical characteristics of calcareous sands used in this study.It should be noted that the studies considered here wereselected due to their similarity in scope, testing procedure,and availability of the initial and physical properties of testedsoils. The data obtained in the experimental part of this studyare presented in Table A1.

Fig. 4 illustrates the grain size distribution curves of all thecalcareous sands. It is clear that the database includes differentsoils with a considerable range of soil particle sizes.

The results obtained from triaxial tests have been presentedin Rezvani (2011) and showed that the two calcareous sandsexhibited different responses to shearing loading. Hormuz Islandsand showed more strength than Bushehr Port sand in similarconditions of relative density and applied confining pressure. Agreater volume change was observed in Bushehr Port sand andthe contractive phase was the major behavior shown duringmost tests on this soil. The responses of the tested sands aredifferent because of various initial properties caused by dissimilar

sand (Hassanlourad et al., 2008).

Fig. 3. Microscopic image of the Kish sand (Hassanlourad et al., 2008).

Table 1Physical properties of the calcareous sands considered in the database.

Name of sand D50 (mm) Gs Cu Cc Dr (%) s03 (kPa) emax emin Particle shape

Ewa Plains, Hawaii a 0.84 2.72 5.42 0.96 39–70.4 50–150 1.30 0.66 Highly angular

Kish Island, the Persian Gulf b 0.51 2.69 3.5 0.9 26–103 50–600 0.72 0.51 Subangular

Hormuz (I) Island, the Persian Gulf b 0.51 2.7 3.6 0.78 23–96 50–600 0.98 0.64 Angular

Rock Beach, England b 0.21 2.72 1.84 1 22–86.5 50–600 1.47 0.84 Needle-shaped and platy

Tonbak (C) Island, the Persian Gulf b 0.44 2.69 3 0.97 24–43 50–600 1.01 0.73 Subangular and platy

Dogs Bay, Ireland c 0.2 2.72 1.92 0.82 61.6–77.2 50–500 2.45 1.62 Angular and platy

Hormuz (II) Island, the Persian Gulf d 0.78 2.76 4.47 0.87 15.1–87 100–600 0.91 0.63 Angular

Bushehr Port, the Persian Gulf d 0.43 2.71 3.2 0.84 27.1–91.4 100–600 1.05 0.73 Subangular

a Morioka and Nicholson (2000).b Hassanlourad et al. (2011).c Hyodo et al. (1994).d Current study.

H. Shahnazari et al. / Computers & Geosciences 50 (2013) 84–94 87

formation conditions. Figs. 5 and 6 show the result of an experi-ment on Hormuz Island sand at confining pressure of 600 kPa.

3. Genetic programming

Soft computing techniques are a series of practical approachesfor solving various types of complicated real-world engineeringproblems, and genetic programming (GP) is one of the most well-known of these methods. GP is a machine learning techniquethat uses optimization and arose from biological reproduction

concepts. Indeed, GP is based on the Darwinian principles ofevolution and natural selection (Koza, 1992). Koza (1992) devel-oped GP from conventional genetic algorithms (GAs) at MIT.

The application of GP to civil engineering problems has beengrowing rapidly among geotechnical researchers in recent years.Some examples include the prediction of foundation settlements(Rezania and Javadi, 2007), modeling the stress–strain behaviorof sand under cyclic loading (Shahnazari et al., 2010) and theprediction of strain energy-based liquefaction resistance of sand-silt mixtures (Baziar et al., 2011). The main advantage of theGP-based techniques (e.g., multigene genetic programming, gene

Fig. 4. The grain size distribution curves of the sands.

Fig. 5. The result of triaxial test on Hormuz sand (II) at 600 kPa confining

pressure: deviatoric stress versus axial strain.

Fig. 6. The result of triaxial test on Hormuz sand (II) at 600 kPa confining

pressure: volume strain versus axial strain.

H. Shahnazari et al. / Computers & Geosciences 50 (2013) 84–9488

expression programming) over regression and other soft computingtechniques is their ability to construct a simplified predictive formula.

In GP, solutions are presented in tree-based structures and arelatively large random population of tree-based individuals witha high level of diversity is generated. Each member of thepopulation consists of functions and terminals that can be chosen

from sets of functions and terminals, respectively. The function setcontains the basic mathematical operators (i.e., addition, multi-plication, etc.) and Boolean logical functions (e.g., AND and OR) orany other user-defined function, while the terminal set contains thenumerical constants and external inputs of the program. A typicalGP flowchart is shown in Fig. 7. More details on genetic program-ming and its operators can be found in Koza (1992).

3.1. Multigene genetic programming (MGP)

A newly developed technique for enhancing the precision ofGP was developed by Hinchliffe et al. (1996) and Hiden (1998)and named multigene genetic programming (MGP). This approachis an improved form of classical GP that uses a new characteristiccalled multigene. In MGP, the model development is based on anumber of genes with non-linear behavior whose combination ina linear form shapes the final structure of the target model. Eq. (1)shows the general form of multigene GP:

Y ¼ a1 � G1þa2 � G2þa3 � G3þ � � � þan � Gnþa0 ð1Þ

where ai is the coefficient of related genes, Gi are the non-lineargenes, n is the number of genes, a0 is the bias term and Y is theoutput. GPTIPS was utilized in the current study to perform aMGP technique for prediction of qmax of calcareous sands. This is anew ‘‘Genetic Programming & Symbolic Regression’’ code devel-oped based on MGP for use with MATLAB (Searson, 2009).

MGP includes some features for setting suitable restrictions toavoid bloating, which causes excessive growth in the model withoutany considerable improvement in fitness value based on the definedfitness function. These restrictions are set by placing limits on someof the configuration parameters, such as the maximum number ofgenes, maximum depth of genes and trees, and maximum numberof nodes per tree. Lexicographic tournament selection introduced byLuke and Panait (2002) was also used for controlling bloating in themodel. This useful technique was utilized by GPTIPS.

The main initial parameters in GPTIPS are maximum numberof genes, maximum length of genes and trees, maximum numberof nodes per tree and tournament selection size, which were usedto set constraints to optimize the modeling process and avoidbloating. The initial parameters of MGP and the variation rangefor each parameter used in this study are shown in Table 2. Otherinitial parameters were set to the default values presented inGPTIPS. More details about MGP can be found in Hinchliffe et al.(1996), Searson (2002 and 2009) and Baziar et al. (2011).

3.2. Gene expression programming (GEP)

Gene expression programming (GEP) is a novel branch of GPthat was first presented by Ferreira (2001). Genetic algorithms(GAs) and genetic programming (GP) are both predecessors ofGEP, in which a computer program (as a problem solution) isencoded in linear chromosomes of fixed length. The chromosomesin the GEP technique are composed of multiple genes, each oneencoding a smaller subprogram. Similar to other evolutionary-based approaches, there are different configuration parameters inGEP that can be employed to optimize the solution and minimizethe error levels, including function set, terminal set, fitnessfunction, control parameters, and termination condition. Table 3illustrates the range of initial parameters used in this study forGEP modeling. The other parameters of GEP-based modeling wereset to their default values according to the GEPSOFT (2006).

The main difference between the MGP and GEP approaches liesin the illustration of the models. In GEP, the models are createdwith a fixed length of character strings (chromosomes) and arefurther described as computer solutions in tree-based structurescalled GEP expression trees (ETs) (Gandomi et al., 2011).

Fig. 7. Typical flow chart of genetic programming (after Koza and Poli, 2005).

Table 2Range of initially defined parameters in MGP.

Parameter Range/setting

Population size 100–10,000

Number of generations 200–5,000

Maximum number of genes 1–15

Maximum number of nodes per tree 3–30

Size of the tournament 2–4

Maximum depth of trees 3–15

Probability of GP tree mutationn 0.1

Probability of GP tree crossovern 0.85

Probability of GP tree direct copyn 0.05

Function set þ , � , � , protected C ,log

n Sum must be equal to 1.

Table 3Range of initially defined parameters in GEP.

Parameter Range/Setting

Number of generations 100–100,000

Number of chromosomes 3–50

Number of genes 1–10

Number of constants per gene 1 QUOTE 7

Numerical constants lower bound �10

Numerical constants upper bound þ10

Function set þ ,� ,� ,C, power, log

Fig. 8. The algorithm of Genetic Expression Programing (Teodorescu and Sherwood,

2008).

H. Shahnazari et al. / Computers & Geosciences 50 (2013) 84–94 89

On the other hand, the models created in MGP are representedin tree-based structures that can be expressed in a functionalprogramming language (Koza, 1992). There is a similar positivefeature between MGP and GEP modeling methods and that istheir multigenic nature which allows the evolution of morecomplex programs composed of several subprograms. Fig. 8

shows the algorithm of the GEP modeling technique. Moredetailed description of the GEP technique can be found inFerreira (2001, 2006), Gandomi et al. (2011), Mollahasani et al.(2011) and Alavi and Gandomi (2011).

3.3. Data division

When modeling with soft computing techniques, it is commonto divide the database into two subsets, training and testing.

Table 4Statistical characteristics of training and testing subsets.

Variable datasets Statistical parameters

Mean Standard deviation Max. Min. Range

Uniformity coefficient, Cu Training 3.51 1.12 5.42 1.84 3.58

Testing 3.68 1.21 5.42 1.84 3.58

All 3.54 1.13 5.42 1.84 3.58

Average size of particles, D50 (mm) Training 0.52 0.21 0.84 0.20 0.64

Testing 0.55 0.22 0.84 0.21 0.63

All 0.53 0.21 0.84 0.20 0.64

Void ratio range, emax�emin or De Training 0.42 0.19 0.83 0.21 0.62

Testing 0.42 0.18 0.64 0.21 0.44

All 0.42 0.19 0.83 0.21 0.62

Confining pressure, s03 (kPa) Training 281.4 196.3 600.0 50.0 550.0

Testing 246.6 189.5 600.0 50.0 550.0

All 274.4 194.4 600.0 50.0 550.0

After consolidation relative density, Drac (%) Training 58.4 25.4 103.0 15.1 87.9

Testing 49.9 23.5 93.0 22.0 71.0

All 56.7 25.1 103.0 15.1 87.9

Maximum deviatoric stress, qpeak (kPa) Training 1025.5 652.2 2465.7 192.0 2273.7

Testing 943.3 631.4 2189.4 149.0 2040.4

All 1009.0 645.4 2465.7 149.0 2316.7

H. Shahnazari et al. / Computers & Geosciences 50 (2013) 84–9490

The training subset is used to calibrate the model and the testingsubset is for validation of the predictive model based on the trainingsubset. The studies performed by Tokar and Johnson (1999) andShahin et al. (2004) have affirmed that the data division method canhave a significant effect on the performance of the models. Shahinet al. (2004) illustrated the impact of data division on the perfor-mance of an ANN-based model in a case study predicting thesettlement of shallow foundations on cohesionless soils. In this study,the database was divided into training and testing subsets in a trialselection procedure, in which the main statistical parameters of thetraining and testing subsets (i.e., maximum, minimum, mean, andstandard deviation) became close to each other. This process deter-mined the likeliest consistent division. Eighty percent of the data(72 cases) were considered for the training subset and twenty percent(18 cases) for the testing subset. Table 4 presents the statisticalcharacteristics of the divided subsets.

4. Results and discussion

4.1. MGP-based formula

To attain simple and straightforward formulae, numerousattempts with different initial settings were performed and theperformance of the developed model was evaluated after each run.All soil parameters cited in Table 1, excluding Cc, which does notincrease the precision of the model, were considered in the modeldevelopment process. Numerical runs also showed that consideringthe void ratio range (emax�emin¼De) as an input parameter resultedin more precise models than the condition in which maximum andminimum void ratios were considered separately in the inputparameters set. The best model was selected according to the beststatistical properties and model simplicity. The statistical parametersconsidered in model evaluation were the coefficient of determination(R2), root mean square error (RMSE) and mean absolute error (MAE).Eq. (2) is the MGP-based formula obtained after simplification.

qpeak ¼ s03102

Cu�68:8

� ��s03D50

155:5

Cuþ119:3D50�190:1

� �

�2635

Dracþ102Deþ228 ð2Þ

The parameters in Eq. (2) and their dimensions are presentedin Table 4. It should be noted that the MGP-based formula

proposed in this study is valid only for the ranges shown inTable 4. Future experiments on the stress–strain of calcareoussands may help to expand the input parameter ranges anddevelop a more comprehensive formula.

Figs. 9–11 compare the MGP-based predictions with the actualvalues of the maximum deviatoric stress of calcareous sands for thetraining, testing and total datasets. The precision of the proposedMGP-based formula was calculated using R2, RMSE and MAE.

The plots illustrated in Figs. 9–11 show that the developedmodel can estimate the maximum deviatoric stress with reason-able precision because all data, including the trained anduntrained cases, are almost completely located within a narrowrange around the angle bisector.

4.2. GEP-based formula

Numerous tests were performed with different initial settingsto obtain an optimal GEP-based model, and the best model wasselected according to its R2, RMSE and MAE. Similarly to the MGP-based model, the best model was obtained without considering Cc

in the input parameters set. Applying De was also more effectivethan the condition in which emax and emin were separatelyconsidered as input parameters. Eq. (3) was found to be the bestGEP-based formula.

qpeak ¼s03

CuDeþ2ðs03þDracÞþD50ð1þDracÞ�

s03Dracð1þCuÞ

þ8:8

D250

þC2u�7:5 ð3Þ

The parameters definitions and ranges in which the GEP-basedformula is valid are presented in Table 4. The precision of thedeveloped equation was examined by plotting the measuredversus predicted values of the maximum deviatoric stress for thetraining, testing, and total datasets as demonstrated in Figs. 12–14,respectively.

As Figs. 12–14 show, most of the cases in the training andtesting subsets were adequately estimated by the GEP-basedmodel. Comparison of the statistical properties that are shownon Figs. 9–14 for MGP and GEP-based models, respectively,illustrates that the MGP-based model is more precise than theGEP-based formula.

Fig. 10. Predicted shearing resistance by MGP-based formula versus measured

values for testing dataset.

Fig. 9. Predicted shearing resistance by MGP-based formula versus measured

values for training dataset.Fig. 11. Predicted shearing resistance by MGP-based formula versus measured

values for all dataset.

H. Shahnazari et al. / Computers & Geosciences 50 (2013) 84–94 91

4.3. Parametric study

A comprehensive parametric study was conducted to evaluatewhether the models’ predictions matched those observed inexperimental investigations. The main goal of this study was todetermine the effect of each input parameter on the value of qmax

in drained conditions. The effect of varying any input variable (inthe range illustrated in Table 4) on the maximum deviatoric stresswas studied while keeping the other parameters constant at theirmean values in the entire database. The results of the parametricstudy on Drac and s03 shown in Figs. 15 and 16 are in agreementwith previous experimental studies (e.g., Dehnavi et al., 2010;Hassanlourad, 2009; Sharma and Ismail, 2006). Relative densityand confining pressure are independent parameters but void ratiorange, uniformity coefficient, and average particle size are

dependent to each other. It means that there are internal relation-ships between Cu, D50, and De and variation of each one canchange the other two parameters simultaneously. In fact, theseparameters are completely dependent to the gradation character-istics of each soil and therefore are not presented here.

4.4. Applying D50–De correlation to the developed models

Cubrinovski and Ishihara (1999) have shown that the void ratiorange (emax�emin or De) is the best parameter for showing the grainsize characteristics of sandy soils, including grain size and grain sizedistribution. They used a set of data from natural deposits of sandysoils and gravels to develop a correlation between D50 and De. Theyalso used data reported by Miura et al. (1997). Fig. 17 illustrates thecorrelation proposed by Cubrinovski and Ishihara (1999) and thedata used in their study. It should be noted that the ranges of D50

and De in the current study’s database (see Table 4) are in the rangesof these parameters illustrated in Fig. 17.

Both of these parameters are input parameters in the modelsdeveloped by the MGP and GEP techniques (Eqs. (2) and (3)). Toobtain other models with fewer input parameters, both modelswere combined with the correlation of D50 and De, and Eqs.(4) and (5) were obtained from the MGP and GEP-based formulae,respectively. To select the best formula, the relative errors (RE) inpredicting qmax were calculated for all the formulae (see Table 5).

qpeak ¼ s03102

Cu�68:8

� ��s03 D50

155:5

Cuþ119:3 D50�190:1

� �

�2635

Dracþ

6:12

D50þ251:5 ð4Þ

qpeak ¼D50s03

Cuð0:23 D50þ0:06Þþ2ðs03þDracÞþD50ð1þDracÞ

�s03

Dracð1þCuÞþ

8:8

D250

þC2u�7:5 ð5Þ

Based on the values in Table 5, the formula developed by theMGP-based technique is more accurate and have a higher rate ofsuccess in predicting the maximum deviatoric stress. The MGP-based formula (Eq. (2)) could predict the qmax of calcareous soilswith less than 10% relative error in more than 75% of the cases in

Fig. 14. Predicted shearing resistance by GEP-based formula versus measured

values for all dataset.

Fig. 12. Predicted shearing resistance by GEP-based formula versus measured

values for training dataset.

Fig. 13. Predicted shearing resistance by GEP-based formula versus measured

values for testing dataset.

Fig. 15. The results of parametric study on MGP and GEP-based formulae for

confining pressure.

H. Shahnazari et al. / Computers & Geosciences 50 (2013) 84–9492

this study’s database. Combining the MGP-based formula withthe correlation proposed by Cubrinovski and Ishihara (1999)which resulted Eq. (4) had less than 10% relative error in 70% ofthe cases. Therefore, Eq. (2) is the best predictive model inconditions for which Cu, D50, Drac, s03 and De are available andEq. (4) is the most appropriate formula in conditions for which Cu,D50, Drac and s03 are accessible.

4.5. Estimation of maximum friction angle using the developed

models

There is a mathematical relationship between the maximumdeviatoric stress and maximum friction angle based on the

principles of soil mechanics that can be represented as follows:

s01s03¼ tan2 45þ

fmax

2

� �ð6Þ

where fmax is in degrees. The definition of maximum deviatoricstress is expressed as:

qmax ¼ s01�s03 ð7Þ

Therefore, s01 can be obtained from Eq. (7), and by substitutingit in Eq. (6) the following relationship is achieved (Eq. (8)):

qmax

s03þ1¼ tan2 45þ

fmax

2

� �ð8Þ

Finally, Eq. (9) is obtained by performing mathematical sim-plifications:

fmax ¼ 2Arc tan

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqmax

s03þ1�90

sð9Þ

In Eq. (9), s03 is one of the available parameters and q can beestimated from the formulae developed in this study.

Table A1Summary of triaxial tests results obtained in this study and used to develop

evolutionary-based models.

Name of sand Dr (%) s03 (kPa) qmax

Bushehr Port, the Persian Gulf a 51.6 100 338.6

27.1 100 285.6

80.1 100 397.9

28.7 200 481.8

58.3 200 572.3

79.2 200 620.2

34.3 400 940.2

56 400 1100.3

88.7 400 1228.7

34.8 600 1419.3

63.2 600 1495.6

91.4 600 1620.6

Hormuz (II) Island, the Persian Gulf a 15.1 100 371.7

42.9 100 395.9

74.8 100 486.8

26 200 676.7

83.5 200 894.6

45.8 200 729.9

28 400 1075.3

40.9 400 1211.1

85.8 400 1382.1

33.9 600 1422.1

87 600 1780.2

41.1 600 1640.3

a Other parameters were presented in Table 1.

Fig. 17. Relationship between void ratio range and mean grain size (Cubrinovski

and Ishihara (1999)).

Fig. 16. The results of parametric study on MGP and GEP-based formulae for

consolidation relative density.

Table 5Relative errors of the developed models.

Predictive model Maximum relative

error percentage

10%

(%)

20%

(%)

30%

(%)

MGP-based formula 75.56 86.67 93.33

GEP-based formula 61.11 83.33 90.00

MGP-based formula developed by relation of

Cubrinovski and Ishihara (1999)

70.00 84.44 92.22

GEP-based formula developed by relation of

Cubrinovski and Ishihara (1999)

53.33 80.00 88.89

H. Shahnazari et al. / Computers & Geosciences 50 (2013) 84–94 93

Consequently, fmax can also be predicted based on the modelsdeveloped in this paper.

5. Conclusions

This paper assembled a database of CD triaxial tests to determinethe maximum deviatoric stress of calcareous soils. The database

consisted of the results presented in the literature along with newresults obtained from the experimental investigations performedduring this study. Two novel predictive models for prediction of theqmax of calcareous sands were developed using two powerful softcomputing techniques, MGP and GEP. The following conclusions canbe made based on the results of this study:

1.

The proposed evolutionary-based models using MGP and GEPmodeling techniques can precisely predict the maximumdeviatoric stress of calcareous sands. Considering their preci-sion and simplicity, the new models can be effectively used inpractice.

2.

The comprehensive parametric study showed adequate per-formance of the models from an engineering perspective. Theresults were also consistent with the findings of previousexperimental studies.

3.

The comparison between the models showed that the MGP-based formula made more precise estimates than the formuladeveloped by the GEP technique. However, both models hadrelatively low error levels.

4.

More applicable formulae with one less input parameter wereobtained by considering a geotechnical relationship from theliterature between two of the input parameters.

5.

A new formula for the estimation of a maximum friction anglebased on the proposed models was obtained by consideringthe relationship between fmax and qmax.

6.

The feasibility of MGP and GEP techniques for finding thesolution of nonlinear problems was demonstrated. Thesemethods can be used to solve other complex engineeringproblems.

Appendix

Table A1 presents 24 triaxial tests data which were obtainedduring the experimental portion of this study.

H. Shahnazari et al. / Computers & Geosciences 50 (2013) 84–9494

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